I would assume that they can hear him at least some of the time.
The 3rd edition rules give the dieties very specific limitations on omniscience, but its still incredibly powerful information gathering ability:
As standard action, a deity of rank 1 or higher can perceive everything within a radius of one mile per rank around any of its worshipers, holy sites, or other objects or locales sacred to the deity. This supernatural effect can also be centered on any place where someone speaks the deity’s name or title for up to 1 hour after the name is spoken, and at any location when an event related to the deity’s portfolio occurs. The remote sensing power can cross planes and penetrate any barrier except a divine shield (described in Salient Divine Abilities) or an area otherwise blocked by a deity of equal or higher rank. Remote sensing is not fooled by misdirection or nondetection or similar spells, and it does not create a magical sensor that other creatures can detect. A deity can extend its senses to two or more remote locations at once (depending on divine rank) and still sense what’s going on nearby...Demigods have a limited ability to sense events involving their portfolios. They automatically sense any event that involves one thousand or more people. The ability is limited to the present. Lesser deities automatically sense any event that involves their portfolios and affects five hundred or more people. Intermediate deities automatically sense any event that involves their portfolios, regardless of the number of people involved. In addition, their senses extend one week into the past for every divine rank they have. Greater deities automatically sense any event that involves their portfolios, regardless of the number of people involved. In addition, their senses extend one week into the past and one week into the future for every divine rank they have. When a deity senses an event, it merely knows that the event is occurring and where it is. The deity receives no sensory information about the event. Once a deity notices an event, it can use its remote sensing power to perceive the event.
So lets assume a Rank 12 intermediate diety. If you mention the diety by name, it can detect it and focus its senses on it to view the event any time in the week after you mention the name. Since they can take 12 standard actions at a time to view events (22 if thier portfolio involves the senses), and it can be assumed dieties spend a reasonable amount of thier time gathering information about events they are going to collect alot of information. If a rank 12 diety spends only 4 hours a day viewing mortals, he'll observe 28800 events a day and observe and understand what happens within 12 miles of each of those events.
Earlier editions did not have so precise of rules, but those seem reasonable if you want to assume something less than omnipotence but still have effective dieties.
I would think that in a world with millions of inhabitants and hundreds of thousands of worshipers, even 28800 events a day might not be enough to bring an individual blasphemer to an intermediate gods attention or concern. Gods are by thier nature very very busy.
Except that your PC has a problem. He's not an ordinary individual. He's not a little peon. He's presumably a fairly high level character who is important locally or globally to the game world and involved in important events likely to attract the interest of the Gods. So I would presume that he's regularly checked up on by every God whose portfolio might be impact by his actions, and by the rules (and logically) the most convienent and interesting time to check up on him is when he mentions the diety by name. Oooops.
And in that case he's screwed. Because if you assume as D&D always has that the gods of D&D are typical members of polytheistic pantheons, then you must assume that they are even the best of the vindictive in the extreme and have very little tolerance for mere mortals doing anything to diminish thier glory. Lèse majesté is not at all tolerated and must be addressed summarily.
At the very least...
a) Any cleric of any of the dieties he's ever cursed will find that thier beneficial spells no longer help him. Any god he's ever cursed will not heal, cure, or ressurect the blasphemer nor suffer thier followers to do the same until the blasphemer has made suitable atonement.
b) Any of the dieties servitors will begin with a hostile attitude toward the character. That could be a problem if the characters ever run across say a Planetar.
c) Any action that the character undertakes which could effect the dieties portfolio will be regarded hostilely.
d) The portofolio of any of the deities he's ever cursed will become hostile to the character. So for example, if he curses a fire god, you may expect that he'll never be able to start a fire under duress, and that any fire he does start (or even gets near) will either mysteriously sputter out (if needed) or else mysteriously spread (if unwanted). If he curses a sea diety, ocean travel will always be disasterous. If he curses a god of horses, then horses will thereafter respond hostilely to his presence, and so forth.
If he's a prominent individual whose behavior could potentially sway others (in earlier editions if he's 9th level or above), then even that would be considered too light of punishment and more direct intervention is likely. I'd give it a 1% chance per incident that the diety would respond immediately, and a greater percent chance if the incident occurred within the dieties sacred sites or places of worship.
As for how I'd handle this...
Normally, I'd warn the character in and possibly out of game the first time this came up. Having a hostile relationship with even a single diety is really putting a heavy burden on yourself, but if the character understood the consequences it could make for interesting RP. However, literally no mortal can afford to have a large number of Gods offended at them. If you follow the above suggestions, you are more or less dooming your player and I would not advocate changing your play style radically because that wouldn't be fair to the player. Instead, assume that the PC is just becoming powerful enough to garner notice and pick one diety as the one that has become offended. If he doesn't take the hint, start adding more. Eventually you may want to approach the player in private to see if this is really the story he wants to create. Enemy of the gods is definately an 'epic destiny' but as Prometheus or Tantalus or even Oddyseus could tell you, it doesn't usually work out well.